Hi Heather! Late April and early May are my favorite times to visit Walt Disney World! We've visited often in those months and have found the crowds are very manageable then. Currently, there don't appear to be any special events scheduled that will drastically affect park attendance during the last two weeks of April. The parks are typically more crowded on the weekends, so you may want to travel to and from Florida on Saturday or Sunday and plan to visit the parks on the weekdays.
When planning my trips, I start by checking the checking the calendars for each park at
disneyworld.com. One good way to avoid crowds is to stay at a Disney resort so that you can take advantage of the Extra Magic Hours, during which time the parks are open only for resort guests. I don't normally like to get up early, but I like crowds even less, so I always plan to be at the Magic Kingdom on the days that resort guests are admitted one hour early. Your young children will love the Fantasyland rides, and if you take advantage of early-entry one morning, you'll find that you can ride most of them with little or no waiting.
I'm always looking for ways to cut costs, but one place I never skimp is on buying the park-hopper. The Magic Kingdom is often quite crowded by lunchtime so we often spend the morning at the Magic Kingdom and then hop over to Epcot for the rest of the day. You might want to try this technique a couple of times during your visit, and then plan to visit the Animal Kingdom and Hollywood Studios on the other days. For those parks, I often plan to visit the one that isn't having Extra Magic Hours in the hope that the crowds will be at the one park that is.
I hope this helps you plan a magical vacation!